On the maximal mean curvature of a smooth surface

TL;DR

This paper investigates the relationship between maximal mean curvature and volume in smooth surfaces, showing that in higher dimensions, small volume can coexist with bounded maximal mean curvature, unlike in planar cases.

Contribution

It demonstrates that unlike planar domains, higher-dimensional smooth embeddings can have arbitrarily small volume despite bounded maximal mean curvature.

Findings

In 2D, area is bounded below by maximal curvature.

In higher dimensions, small volume is possible with bounded maximal mean curvature.

Abstract

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.